A converse to a theorem of Adamyan, Arov and Krein

J.~Agler and N.~J.~Young

Keywords

interpolation, meromorphic function, pseudomultiplier

Status

J. Amer. Math. Soc. 12(1999) 305--333.

Abstract

A well known theorem of Akhiezer, Adamyan, Arov and Krein gives a criterion (in terms of the signature of a certain Hermitian matrix) for interpolation by a meromorphic function in the unit disc with at most $m$ poles subject to an $L^\infty$-norm bound on the unit circle. One can view this theorem as an assertion about the Hardy space $H^2$ of analytic functions on the disc and its reproducing kernel. A similar assertion makes sense (though it is not usually true) for an {\em arbitrary} Hilbert space of functions. One can therefore ask for which spaces the assertion {\em is} true. We answer this question by showing that it holds precisely for a class of spaces closely related to $H^2$. \

Nicholas Young
28 October 1999
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